2. RINGS AS COLLISIONAL OBJECTS

The major breakthrough in our understanding of the curious ring
galaxies came from two sources, namely the work of
Lynds and Toomre (1976),
and Theys and Spiegel
(1977).
In both cases ring galaxies were
hypothesized to be the result of a head-on collision between a compact
companion galaxy and a larger disk system. The resulting gravitational
perturbation was shown to generate rings in the disk of the larger
system. The backdrop to this rather startling idea was the earlier
pioneering work on tidal interactions of
Toomre and Toomre
(1972)
who had already begun to destroy the myth of the permanence of large scale
galactic structure over a Hubble time. A common feature of the
collisional ring formation picture is the generation of radially
expanding density waves resulting from the crowding of stars in the
disk. (A more complete description of the kinematics and dynamics is
presented in Section 4.) Although the most
coherent effects are likely
to be found in the dynamically cool disk stars and gas, the central
perturbation will also have consequences for the halo stars and any
dark matter present.

Lynds and Toomre (1976)
first presented the elegant conceptual model
that forms the basis of our understanding of the ring galaxy
phenomenon. In the simplest form of this model, a small companion
galaxy is assumed to pass down the symmetry axis of the larger primary
galaxy, and move through the disk center. Prior to the collision the
stars in the primary disk are assumed to be in circular orbits. At the
time of impact the disk stars feel a strong pull toward the center as
a result of the companion's gravity. (Note: the cross section of each
star is so small that virtually all of the stars in the two galaxies
will pass by each other in the "collision"). In the simplest case we
also assume that the collision is so rapid that the disk stars do not
have time to adjust to the sudden inward impulse. Specifically, in the
so-called impulse approximation (IA), the stellar positions are
assumed to be the same immediately before and after the collision, but
after the collision each star has acquired an inward radial velocity
(see e.g.,
Tremaine 1981).
As long as the perturbation is not too
large, the resulting stellar orbits in the disk can be well
approximated by radial epicyclic oscillations about a guiding center,
i.e., the precollision orbit. Lynds and Toomre graphically
illustrated this with the example of planetary orbits following a
hypothetical near collision between the Sun and another star. This
solar system example also reminds us of the ancient origin of
epicycles as the simplest modification of "perfect" circular orbits.

Once we assume that the perturbing force only affects the disk stars
for a short time, then subsequent motions are purely kinematic.
Specifically, this kinematic approximation neglects the effects
of the self-gravity of the perturbation. It is especially appropriate for
collisions with large relative velocities (e.g., such as probably
occur in galaxy clusters). The kinematic approximation assumes a
decoupling of the perturbation from the resulting motions without
specifying how the perturbation is derived.

After the companion passes through, the disk, individual stars begin
their epicyclic oscillations. In general, the period of these
oscillations increases with radius throughout most of the disk. For
example, in a flat rotation curve disk the epicyclic frequency scales
as v/r, so the period scales as
Pr. Thus, while the stars at
a given radius have rebounded and begun to move outwards, those at a
slightly larger radius are still moving inwards. The consequence of
this radial dispersion is that stellar orbits will bunch or crowd at
some radii, yielding high densities there (see
Toomre 1978).
At other
radii the orbits spread, giving rise to rarefied regions. These
effects are well demonstrated by radius versus time diagrams.
Figure 3
shows the first such used for ring galaxies by
Toomre (1978).
The region of orbit crowding propagates outward as a density wave.

Figure 3. Radial trajectories of 40
particles from the symmetrical encounter model of
Toomre (1978).
The crowding of the particles into
the "rings" is clearly demonstrated in this pioneering model.

Figure 3 illustrates several other important
points. In the first
ring the orbit crowding occurs almost exclusively among particles
rebounding outward. It also shows that a determination of the outflow
velocity of an individual star or HII region in some part of the first
ring probably gives a good indication of the outflow of all of the
material in that region. In the second ring, infalling and outflowing
stars cross each other in a sharply defined high-density region with
sharp caustic edges. This is a qualitatively different behavior, and
will be further examined in Section 4. In
this case, determining ring
propagation speeds from individual stellar, HII region or gas cloud
radial velocities is much more problematic.

The orbit crossing zones become wider in later rings, and they
overlap each other. This radial phase mixing, together with the fact
that the later rings include ever fewer stars, eventually renders the
rings indistinguishable and ultimately invisible. There are a couple
of caveats, however. First, even at very late times, the simple models
indicate that individual rings can separate out at large
radii. Secondly, even when the rings overlap, some memory of the
collision is retained in the continuity of epicyclic phases with
radius. Eventually collisional diffusion, through interactions between
stars and massive molecular clouds, will erase this memory too.

Thus far, we have only considered perturbations to stellar orbits
within the initial plane of the primary disk. This is justified
because the perturbations perpendicular to the disk plane
(i.e. in the
"z" direction) are second order in the radial perturbation amplitude
in the impulse approximation. The companion pulls the stars upward in
the z direction when it approaches from above, and downward after it
passes through. On the other hand, in-plane radial impulses are
first order. This is because the companion pulls disk stars radially
inward, towards the galaxy's center, when it is both above and below
the plane. Yet in the z-direction, even second order effects can be
important, especially when the companion is massive and moving
slowly. Numerical simulations show several very interesting
effects. One example is a vertical tidal effect that was well
illustrated by
Lynds and Toomre's
(1976)
Figure 5, reproduced in
Figure 4. This figure shows that, as the
companion approaches, stars
in the central regions of the primary are pulled up towards it more
that stars in the outer disk. Similarly, once the companion passes
through the primary, stars near the center follow it downward while
the outer disk stars are still moving upwards. Viewed edge-on the
outer disk appears to nap like a bird's wings relative to the primary
center. In conventional terms this leads us to expect a significant
warping of the disk in any relatively young ring galaxy. Indeed,
there is evidence from simulations (Appleton and James, unpublished)
that the flapping can become so vigorous that stars in the outermost
disk can be shaken off.

Figure 4. The first numerical simulations
of collisional ring galaxies
(Lynds and Toomre,
1976).
The companion mass is 2/3 of the target
mass, and "time is reckoned in units of
(r03 / GM)1/2", where
M is the target mass, and the scale length r0
is shown.